PICSL : Semi-Lagrangian and Particle Methods for Solving the Vlasov Equation Y. Barsamian, J. Bernier, S. Hirstoaga, M. Mehrenberger, P. Navaro Universities of Rennes & Strasbourg August 2016 PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 1 / 20
Outline What is plasma ? How can we model its dynamics ? How can we code a simulation in the chosen model ? How can we optimize that code ? PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 2 / 20
Examples of plasma The fourth state of matter. . . 99% of the universe ! PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 3 / 20
Examples of plasma The fourth state of matter. . . 99% of the universe ! lightning PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 3 / 20
Examples of plasma The fourth state of matter. . . 99% of the universe ! fluorescent light PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 3 / 20
Examples of plasma The fourth state of matter. . . 99% of the universe ! ITER a tokamak (controlled thermonuclear fusion) ≪ The way ≫ (in latin) to produce energy a . PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 3 / 20
Modelling ∂ f v · ∂ f x − e − → E · ∂ f ∂ t + − → v = 0 Vlasov ∂ − → ∂ − → m − ∆ φ = ρ Poisson ε 0 f ( − → x , − → v , t ) : distribution function of the electrons − → x , t ) = − − − → E ( − → grad φ : the electric field, here self-induced ; φ is the associated scalar potential ε 0 : vacuum permittivity e , m : electron charge and mass t : time − → x ∈ ( R / ( L x Z )) × ( R / ( L y Z )) : particle position (1d, 2d or 3d) → − v ∈ R 2 : particle speed (1d, 2d or 3d) � � � ρ ( − → f ( − → x , − → v , t ) d − → 1 − x , t ) = e v : volume charge density PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 4 / 20
Modelling Essentially three methods for modelling the particle density inside plasma : Semi-Lagrangian methods Particle-in-Cell methods Eulerian methods PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 5 / 20
Semi-Lagrangian Methods splitting of the Vlasov equation → − ∂ f v · ∂ f x + q E · ∂ f ∂ t + − → v = 0 ∂ − → ∂ − → m PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 6 / 20
Semi-Lagrangian Methods splitting of the Vlasov equation into two simpler equations : ∂ f v · ∂ f − → E · ∂ f ∂ f v · ∂ f x + q ∂ t + − → ∂ t + − → v = 0 x = 0 ∂ − → ∂ − → ∂ − → m − → ∂ f ∂ t + q E · ∂ f [Cheng and Knorr, 1976] v = 0 ∂ − → m PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 6 / 20
Semi-Lagrangian Methods splitting of the Vlasov equation into two simpler equations : ∂ f v · ∂ f → − E · ∂ f ∂ f v · ∂ f x + q ∂ t + − → ∂ t + − → v = 0 x = 0 ∂ − → ∂ − → ∂ − → m − → ∂ f ∂ t + q E · ∂ f [Cheng and Knorr, 1976] v = 0 ∂ − → m follow the characteristics : x x Values after k time steps. Values after k + 1 time steps. PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 6 / 20
Semi-Lagrangian Methods splitting of the Vlasov equation into two simpler equations : ∂ f v · ∂ f → − E · ∂ f ∂ f v · ∂ f x + q ∂ t + − → ∂ t + − → v = 0 x = 0 ∂ − → ∂ − → ∂ − → m − → ∂ f ∂ t + q E · ∂ f [Cheng and Knorr, 1976] v = 0 ∂ − → m follow the characteristics : g ∗ ( x , ( k + 1) δ t ) x x Values after k time steps. Values after k + 1 time steps. PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 6 / 20
Semi-Lagrangian Methods splitting of the Vlasov equation into two simpler equations : ∂ f v · ∂ f − → E · ∂ f ∂ f v · ∂ f x + q ∂ t + − → ∂ t + − → v = 0 x = 0 ∂ − → ∂ − → ∂ − → m − → ∂ f ∂ t + q E · ∂ f [Cheng and Knorr, 1976] v = 0 ∂ − → m follow the characteristics : Advection g ∗ ( x , ( k + 1) δ t ) g ( x − a δ t , k δ t ) x x Values after k time steps. Values after k + 1 time steps. PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 6 / 20
Semi-Lagrangian Methods splitting of the Vlasov equation into two simpler equations : ∂ f v · ∂ f − → E · ∂ f ∂ f v · ∂ f x + q ∂ t + − → ∂ t + − → v = 0 x = 0 ∂ − → ∂ − → ∂ − → m − → ∂ f ∂ t + q E · ∂ f [Cheng and Knorr, 1976] v = 0 ∂ − → m follow the characteristics : Interpolation g ( x − a δ t , k δ t ) x x Values after k time steps. Values after k + 1 time steps. PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 6 / 20
Particle-in-Cell Methods approximation of f via (a lot of) numerical particles one numerical particle represents many real-life particles PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 7 / 20
Particle-in-Cell Methods approximation of f via (a lot of) numerical particles one numerical particle represents many real-life particles particles only interact via the self-induced fields (but don’t consider every interaction - it’s not N-body model) PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 7 / 20
Particle-in-Cell Methods approximation of f via (a lot of) numerical particles one numerical particle represents many real-life particles particles only interact via the self-induced fields (but don’t consider every interaction - it’s not N-body model) � N f ( − → x , − → weight k δ ( − → x − − x k ) δ ( − → → v − − → v , t ) = v k ) k =1 δ is the distribution of Dirac : � δ ( x ) dx = 1 R δ (0) = + ∞ δ ( x ) = 0 when x � = 0 PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 7 / 20
Comparison PIC / SL Methods SL + only stores f via a grid on positions and speeds : faster on 1D (2D grid) and 2D (4D grid) - slower on 3D (6D grid is too much) PIC + only stores a grid for the fields on positions : faster on 3D - also stores an array of particles : slower on 1D and 2D 1 - requires a lot of particles : stochastic convergence in √ N PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 8 / 20
Dispersion relation Principle : solve exactly the linearized equation. If f 0 ≡ f 0 ( v ) is an equilibrium solution and f (0 , v ) e ik . x where A ≪ 1 then f ( t = 0 , x , v ) = f 0 ( v ) + A � � Res ( ω ) e − i ω t k E ( t , x ) = Ae ik . x | k | + O ( A 2 ) , ω ∈ D − 1 ( { 0 } ) with D an analytic function depending only on f 0 and k and Res an analytic function depending on f 0 , k and � f (0 , . ). ∀ ω ∈ D − 1 ( { 0 } ) , I m( ω ) ≤ 0 ⇒ stable. e.g. f 0 = e − v 2 2 √ . 2 π ∃ ω ∈ D − 1 ( { 0 } ) , I m( ω ) > 0 ⇒ unstable. e.g. f 0 = v 2 e − v 2 2 √ . 2 π PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 9 / 20
Test case : Landau f (0 , x , v ) = e − v 2 2 (1 + A cos( kx )) ⇒ ∀ ω ∈ D − 1 ( { 0 } ) , I m( ω ) ≤ 0 √ 2 π -2 Simulated solution (SL) -3 0.5 log(Electric energy) Linearized solution - first time mode -4 -5 -6 -7 -8 -9 -10 -11 -12 0 2 4 6 8 10 12 14 16 18 Time (adimensionned) PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 10 / 20
Test case : 2 d × 2 d Problem : linearly, 2 d solution is a superposition of 1 d solutions. Solution : find the term in A 2 in the expansion of f . Principle : if f = f 0 ( v ) + Af 1 ( t , x , v ) + A 2 f 2 ( t , x , v ) and E = AE 1 ( t , x ) + A 2 E 2 ( t , x ) then ∂ t f 2 + v . ∇ x f 2 − E 2 . ∇ v f 0 − E 1 . ∇ v f 1 = 0 , � − ∆ x Φ 2 = − R 2 f 2 dv , E 2 = −∇ x Φ 2 . It is the linearized equation but with a source term E 1 . ∇ v f 1 that is given by the linear analysis. The solution is given by the Duhamel’s formula. Consequence : one can deduce the dominant time mode of f 2 . PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 11 / 20
Test case : 2 d × 2 d f ( t = 0 , x , v ) = f 0 ( v ) + A α ( v ) e ik 1 . x + A β ( v ) e ik 2 . x k 1 , ω 1 k 2 , ω 2 PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 12 / 20
Test case : 2 d × 2 d f ( t = 0 , x , v ) = f 0 ( v ) + A α ( v ) e ik 1 . x + A β ( v ) e ik 2 . x 2 k 1 , { 2 ω 1 } ∪ D − 1 2 k 1 ( { 0 } ) k 1 , ω 1 k 1 + k 2 , { ω 1 + ω 2 } ∪ D − 1 k 1 + k 2 ( { 0 } ) k 2 , ω 2 2 k 2 , { 2 ω 2 } ∪ D − 1 2 k 2 ( { 0 } ) PICSL Project (Rennes / Strasbourg) Plasma physics via computer simulation 25/08/2016 12 / 20
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